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Asymptotic analysis of the autocorrelation function of
the wavelet packet coefficients of a band-limited
wide-sense stationary random process
Abdourrahmane M. Atto, Dominique Pastor, Alexandru Isar
To cite this version:
Abdourrahmane M. Atto, Dominique Pastor, Alexandru Isar. Asymptotic analysis of the autocorrela-tion funcautocorrela-tion of the wavelet packet coefficients of a band-limited wide-sense staautocorrela-tionary random process. [Research Report] Dépt. Signal et Communications (Institut Mines-Télécom-Télécom Bretagne-UEB); Politehnica University Timisoara (University of Timisoara). 2006, pp.36. �hal-02316459�
Asymptotic Analysis of the Autocorrelation
Function of the Wavelet Packet Coefficients of
a Band-Limited Wide-Sense Stationary
Random Process
Analyse asymptotique de la fonction
d’autocorr´elation des coefficients des paquets
d’ondelettes associ´es `
a un processus al´eatoire
stationnaire au sens large et `
a bande limit´ee
Rapport Interne GET / ENST Bretagne
Abdourrahmane M. Atto, Dominique Pastor
Ecole Nationale Sup´erieure des T´el´ecommunications de Bretagne,
Technopˆole de Brest Iroise, CS 83818,
29238 BREST Cedex,
FRANCE
e-mail:
{am.atto, dominique.pastor} @enst-bretagne.fr
&
Alexandru Isar,
Univ. ’Politehnica’ din Timisoara,
Bd. Vasile Parvan, nr. 2,
300223 Timisoara,
ROMANIA
Abstract
This paper is a contribution to the analysis of the statistical correlation of the wavelet packet coefficients issued from the decomposition of a random pro-cess, stationary in the wide-sense, whose power spectral density is bounded with support in [−π, π].
Consider two quadrature mirror filters (QMF) that depend on a param-eter r, such that these filters tend almost everywhere to the Shannon QMF when r increases. The parameter r is called the order of the QMF under consideration. The order of the Daubechies filters (resp. the Battle-Lemari´e filters) is the number of vanishing moments of the wavelet function (resp. the spline order of the scaling function).
Given any decomposition path in the wavelet packet tree, the wavelet packet coefficients are proved to decorrelate for every packet associated to a large enough resolution level, provided that the QMF order is large enough and above a value that depends on this wavelet packet.
Another consequence of our derivation is that, when the coefficients asso-ciated to a given wavelet packet are approximately decorrelated, the value of the autocorrelation function of these coefficients at lag 0 is close to the value taken by the power spectral density of the decomposed process at a specific point. This specific point depends on the path followed in the wavelet packet tree to attain the wavelet packet under consideration.
Some simulations highlight the good quality of the “whitening” effect that can be obtained in practical cases.
Keywords
Wavelet packets, wide-sense stationary random process, autocorrelation func-tion, Daubechies filters, Battle-Lemari´e filters, mean-square integral, spectral representation of a process.
R´
esum´
e
Dans ce papier, on analyse la corr´elation statistique des coefficients des pa-quets d’ondelettes associ´es `a un processus al´eatoire, stationnaire au sens large et dont la densit´e spectrale est `a support dans [−π, π].
Soient deux filtres miroirs en quadrature (FMQ) d´ependant d’un param`etre r tel que ces filtres tendent presque partout vers les FMQ id´eaux de Shannon lorsque r croˆıt. Le parameter r est appel´e ordre des filtres FMQ consid´er´es. L’ordre des filtres de Daubechies est le nombre de moments nuls de la fonc-tion d’ondelette. L’ordre des filtres de Battle-Lemari´e est l’ordre de la spline associ´ee `a la fonction d’´echelle.
Etant donn´e un chemin de l’arbre de d´ecomposition en paquets d’ondelettes, nous montrons que les coefficients des paquets d’ondelettes tendent `a se d´ecorr´eler pour chaque paquet associ´e `a un niveau de r´esolution suffisament grand, `a condition que l’ordre des FMQ soit lui-aussi suffisament grand et sup´erieur `a une valeur qui d´epend du paquet d’ondelettes consid´er´e.
Un autre cons´equence de ce r´esultat est le suivant: lorsque les coeffi-cients associ´es `a un paquet d’ondelettes sont approcimativement d´ecorr´el´es, la valeur de la fonction d’autocorr´elation en 0 est proche de la valeur de la densit´e spectrale du processus en un point que l’on sait d´eterminer. Ce point d´epend du chemin suivi dans l’arbre de d´ecomposition pour atteindre le paquet d’ondelettes consid´er´e.
Quelques simulations mettent en ´evidence la bonne qualit´e de l’effet de blanchiment que l’on obtient en pratique.
Mots-cl´
es
Paquet d’ondelettes, processus stationnaire au sens large, fonction d’autocorr´elation, filtres de Daubechies, filtres de Battle-Lemari´e, int´egrale en moyenne
Contents
1 Introduction 5
2 Discrete wavelet packet decomposition of a wide-sense
sta-tionary random process 7
2.1 Discrete wavelet packet analysis . . . 7 2.2 Discrete wavelet packet decomposition of a second-order WSS
random process . . . 9 3 Asymptotic decorrelation of the wavelet packet coefficients
of a band-limited wide-sense stationary random process 11 3.1 The binary sequence associated to a given sequence of wavelet
packets . . . 13 3.2 Asymptotic decorrelation with the Shannon wavelet packets . 14 3.3 Asymptotic decorrelation with non-ideal QMF . . . 19
4 Experimental results 21
4.1 Influence of the resolution level and the order of the filters on the decorrelation process . . . 22 4.2 The limit value of the correlation function at lag 0 . . . 23
5 Conclusion 23 Appendix A 31 Appendix B 31 Appendix C 32 Appendix D 33 Appendix E 34
1
Introduction
The transforms that map an original continuous-time random process into a set of uncorrelated random variables can be regarded as optimal discreti-sations in the sense that the analysis of a random sequence whose elements are not correlated is much simpler than the analysis of a continuous-time random process. Therefore, such transforms are of much practical interest in signal processing and communications applications. The Karhunen-Lo`eve (KL) expansion is a typical example of such an optimal discretisation. It applies to any Wide-Sense-Stationary (WSS) random process, that is any Hilbertian or second-order process X(t) (E[|X(t)|2] < ∞) whose correlation function R(t, s) = E[X(t)X(s)] depends only on the time-increment t − s between its two arguments.
The analysis of time-series through the Wavelet transform has then re-ceived much interest in the recent years. Many authors have studied various aspects of the statistical correlation of the wavelet coefficients (see [1–7], amongst others). These works highlight that, for many stationary and non-stationary input random processes, the between-scale and within-scale coeffi-cients returned by a Discrete Wavelet Transform (DWT) tend to decorrelate as the resolution level increases; for further details, the reader may refer to the very nice overview given in [7]. The DWT can thus be regarded as a “nearly” optimal discretisation. For WSS processes, it is a relevant alternative to the KL theory because, unlike the KL expansion, the DWT expansion achieves the “whitening” effect without solving any eigen equation and without hav-ing to assume that the process is time-limited, a constraint of importance in the KL theory. For instance, in [8], the DWT “whitening effect” serves to design a telecommunication receiver robust in presence of WSS noise; this system performs without knowing the noise Power Spectral Density (PSD) and it yields Binary Error Rates comparable to those achieved by the op-timal receiver based on the noise KL expansion. Note also that when the autocorrelation function of a given WSS process is known, it is even possible to construct a non-orthogonal wavelet basis in terms of which the process can be expanded with uncorrelated coefficients; the construction of this basis still requires no eigen equation to solve and does not need assume that the process is time-limited [9].
In complement with the results recalled above and dedicated to the DWT, the present paper addresses the Discrete Wavelet Packet Transform (DWPT) of a centred WSS random process. In particular, it proposes an analysis of the “whitening” effect that the DWPT can achieve in any path when the resolution level increases. This analysis is motivated by the following facts.
that are returned at node (j, n) of the wavelet packet tree by decomposing the input WSS random process; j is the resolution level and n is the shift parameter valued in {0, 1, . . . , 2j − 1} (see section 2.2).
If n is constant with j or if the value of n depends on j but remains upper-bounded by a constant independent of j, it follows from [10] that, when j tends to infinity, the coefficients (cj,n[k])k∈Z tend to decorrelate and that the autocorrelation function at lag 0 of the discrete random process (cj,n[k])k∈Z tends to the value of the PSD of the input random process at the origin.
However, as explained below, when n is not a fixed constant or does not remain upper-bounded by a value independent of j, the analysis of the autocorrelation function of (cj,n[k])k∈Z becomes significantly more intricate: In this case, given an arbitrary pair of quadrature mirror filters (QMF), we cannot guarantee that increasing only the resolution level is sufficient to obtain asymptotically decorrelated wavelet packet coefficients. Moreover, even when the “whitening” effect is guaranteed, the value of the asymptotic autocorrelation function at lag 0 of the discrete random process (cj,n[k])k∈Z is no longer the value of the PSD at the origin but actually depends on the sequence of the values of n in the DWPT path followed when j varies; for example, if we choose n = 2j− 1 at every resolution level j, the value of the asymptotic autocorrelation function at the origin is the value taken by the PSD at π.
The principal contribution of this paper is then theorem 1. The asymp-totic decorrelation of the wavelet packet coefficients stated by this theorem for a large class of WSS random processes met in practice is obtained by considering QMF h[r]0 and h[r]1 whose Fourier transforms H0[r] and H1[r] tend almost everywhere to the Fourier transforms of the Shannon QMF when r increases. The parameter r is hereafter called the order of the QMF h[r]0 and h[r]1 . If h[r]0 and h[r]1 are Daubechies QMF, the order r is the number of van-ishing moments of the wavelet function associated to H0[r]; if these QMF are Battle-Lemari´e filters, the order r is the spline order of the scaling function associated to H0[r].
Several papers already stressed the importance of parameters such as the order r for analysing the statistical correlation of the DWT and DWPT coefficients. For a fractional Brownian motion, [1] shows that the larger the number of the wavelet vanishing moments, the more decorrelated the wavelet coefficients. In [11], an asymptotic within-scale nearly-whiteness inequality is achieved for the wavelet packet coefficients when the regularity of the scaling function increases. Recently, [7] highlighted the role played by the length L of the impulse response of the Daubechies filters to obtain decorrelated between-scale wavelet coefficients: The covariance of the between-between-scale coefficients of
some stationnary process tends to 0 at rate L−1/4 as L tends to infinity. This paper is organized as follows. In section 2, the reader is reminded with basic results concerning the wavelet packet decomposition of a random process. In particular, we will give the expressions of the autocorrelation functions of the discrete sequences formed by the wavelet packet coefficients. The convergence of these function sequences when the resolution level tends to infinity is studied in section 3, and is achieved in two steps. In section 3.2, the asymptotic decorrelation is established for the Shannon wavelet packet decomposition, which employs the Shannon QMF that are ideal filters. In section 3.3, we consider QMF for which the above notion of order makes sense. We use the convergence of such filters to the Shannon filters when the order increases to prove that the wavelet packet coefficients tend to be decorrelated when both the resolution level and the order of the QMF are large enough; the order must be chosen according to the resolution level. In section 4, we present some experimental results to illustrate the role played by the resolution level and the order of the QMF in the decorrelation process. Finally, we conclude in section 5.
2
Discrete wavelet packet decomposition of a
wide-sense stationary random process
In section 2.1, we present some aspects concerning the wavelet packet analy-sis. We adopt the same notations as [12]. Section 2.2 gives the decomposition of a WSS random process and the expressions of the autocorrelation functions of its wavelet packet coefficients.
2.1
Discrete wavelet packet analysis
Let Φ be a function such that {τkΦ : k ∈ Z} is an orthonormal system of L2(R), where τkΦ : t 7−→ Φ(t−k). Let U be the closure of the space spanned by this orthonormal system.
Consider two QMF h0 and h1 and define H0(ω) = √1 2 P `∈Zh0[`]e−i`ω H1(ω) = √1 2 P `∈Zh1[`]e−i`ω. (1)
The functions H0 and H1 are respectively the Fourier transforms of h0 and h1 (up to the factor 1/√2). The quadrature mirror condition is equivalent
to the unitarity of the matrix M (ω) = H0(ω) H1(ω) H0(ω + π) H1(ω + π) , (2)
for every ω. We assume that the functions H0 and H1 are such that
H1(ω) = e−iωH0(ω + π). (3) where z stands for the complex conjugate of z.
We define the sequence (Wn)n≥0 of elements of L2(R) by : W0(t) = √2P`∈Zh0[`]Φ(2t − `) W1(t) =√2P`∈Zh1[`]Φ(2t − `), (4)
and by setting, for all n ≥ 1, W2n(t) = √2P`∈Zh0[`]Wn(2t − `) W2n+1(t) = √2P`∈Zh1[`]Wn(2t − `). (5)
The wavelet packet functions are then defined by
Wj,n(t) = 2−j/2Wn(2−jt). (6) For j ≥ 1 and k ∈ Z, we put Wj,n,k= τ2jkWj,n, that is
Wj,n,k(t) = 2−j/2Wn(2−jt − k). (7) The set {Wj,n,k : k ∈ Z} is orthonormal. With a slight abuse of language, the closure of the space spanned by {Wj,n,k : k ∈ Z} will hereafter be called the packet Wj,n.
The wavelet packet decomposition of the function space U is obtained by recursively applying the so-called splitting lemma (see [13], for example). We thus can write that
U = W1,0⊕ W1,1, (8) and
Wj,n= Wj+1,2n⊕ Wj+1,2n+1, (9) for every j = 1, 2, · · · , and every n ∈ Ij with
U hhhh hhh ( ( ( ( ( ( ( W1,0 ```` W2,0 aa ! ! W3,0 W3,1 W2,1 aa ! ! W3,2 W3,3 W1,1 ```` W2,2 aa ! ! W3,4 W3,5 W2,3 aa ! ! W3,6 W3,7
Figure 1: Wavelet packet decomposition tree down to resolution level j = 3.
The sets {Wj+1,2n,k : k ∈ Z} and {Wj+1,2n+1,k : k ∈ Z} are orthonormal bases of the vector spaces Wj+1,2n and Wj+1,2n+1, respectively. The wavelet packet tree of figure 1 illustrates such a decomposition
The projection of a function f ∈ U on Wj,n yields the coefficients cj,n[k] =
Z R
f (t)Wj,n,k(t)dt. (11)
Remark 1 With respect to what follows, it is worth emphasizing that the decomposition concerns an arbitrary space U generated by the translated ver-sions of Φ. Therefore, the function Φ is not necessarily the scaling function associated to the low-pass filter H0. If Φ is this scaling function, we have W0 = Φ in (4).
In practice, when the input data of the wavelet packet decomposition are the samples of some function that satisfies Shannon’s sampling theorem, we implicitly use the wavelet packet decomposition of the space U = US of those elements of L2(R) that have their Fourier transforms with support in [−π, π]. The elements of US are said to be band-limited functions. According to Shan-non’s sampling theorem, US is the space spanned by the translated versions of ΦS = sinc.
2.2
Discrete wavelet packet decomposition of a
second-order WSS random process
Let (Ω, A, P ) be a probability space, where P is a probability measure on the elements of A, and let X : R × Ω −→ R be a second-order random process:
kXk2L2(Ω)= E |X|2= Z Ω|X| 2 dP < ∞. (12) We assume that X is centred and continuous in quadratic mean. The au-tocorrelation function of X, denoted by R, is a continuous function. This function is defined by
The wavelet packet decomposition of X returns, at level j and for n ∈ Ij, the random variables
cj,n[k] = Z
R
X(t)Wj,n,k(t)dt, k ∈ Z, (13) provided that the Riemann integral
ZZ R2
R(t, s)Wj,n,k(t)Wj,n,k(s)dtds (14)
exists (see appendix A). The discrete random process (cj,n[k])k∈Z, where cj,n[k] is defined by the mean-square integral (13), represents the sequence of the coefficients of X on the packet Wj,n.
Remark 2 according to the splitting lemma (9), Wj,n is decomposed into two packets Wj+1,2n and Wj+1,2n+1. For ∈ {0, 1}, the coefficients of X on the packet Wj+1,2n+ are defined by
cj+1,2n+[k] = Z
R
X(t)Wj+1,2n+,k(t)dt. (15)
We have (see appendix B),
Wj+1,2n+,k =X `∈Z
h[` − 2k]Wj,n,` (16) and then, we can write, with convergence in L2(Ω),
cj+1,2n+[k] =X `∈Z
h[` − 2k]cj,n[`]. (17) Thus, we can obtain the wavelet packet coefficients of a second-order random process by applying Mallat’s algorithm based on convolution and downsam-pling [14]. The principle of the decomposition is then the same as that used to decompose functions of L2(R), the only difference being the kind of con-vergence involved.
Now, let Rj,nstand for the autocorrelation function of the discrete random process cj,n defined by (13). We have
Rj,n[k, `] = Ecj,n[k]cj,n[`] = E Z R X(t)Wj,n,k(t)dt Z R X(s)Wj,n,`(s)ds = Z Z R2 R(t, s)Wj,n,k(t)Wj,n,`(s)dtds. (18)
Assume that X is a WSS random process. As usual, we write that R(t, s) = R(t − s). We also assume that X has a Power Spectral Density (PSD) γ, which is the Fourier transform of R, and that γ ∈ L∞(R).
If we denote by bf the Fourier transform of any element f of L1(R) or L2(R), it follows from (18) and appendix C, that the discrete random process cj,n is WSS and that Rj,n[m] = 1 2π Z R γ(ω 2j)| cWn(ω)| 2eimωdω. (19)
where, with the same language abuse as above, Rj,n[k − `] = Rj,n[k, `]. This autocorrelation function can also be written
Rj,n[m] = 1 2π Z R γ(ω)|2j/2Wcn(2jω)|2ei2 jmω dω. (20)
With an easy change of variable, we obtain that
Rj,n[m] = 1 2π
Z R
γ(ω)| dWj,n(ω)|2ei2jmωdω, (21) which will prove useful in the sequel.
3
Asymptotic decorrelation of the wavelet packet
coefficients of a band-limited wide-sense
sta-tionary random process
Our purpose is to analyse the behaviour of the autocorrelation functions (21) when the resolution level j tends to infinity.
If n is constant, Lebesgue’s dominated convergence theorem can be used to compute the limit of Rj,n[m] when j tends to infinity. More precisely, if we assume that the PSD γ of X is an element of L∞(R), it follows from (19) that
|Rj,n[m]| ≤ kγk∞kWnk2L2(R) = kγk∞, (22)
and therefore, if n is constant, we have from (19) that :
lim j→+∞Rj,n[m] = 1 2πγ(0) Z R |cWn(ω)|2eimωdω. (23) But Z R |cWn(ω)|2eimωdω = 2π hτmWn, Wni = 2πδ[m], (24)
where δ[m] = 1 if m = 0 0 if m 6= 0 (25) and then lim j→+∞Rj,n[m] = γ(0)δ[m]. (26) The result thus obtained is that given in [10] and is partially proved in [15]. Note that (26) embraces the cases n = 0 and n = 1, which respectively correspond to the approximation coefficients and the detail coefficients of the DWT.
The situation becomes more intricate if n is a function of j. For instance, if we choose n = 2j− 1 for all j > 0 or n = 2j−L where L ∈ N, the behaviour of Rj,n[m] when j tends to infinity is no longer a straightforward consequence of Lebesgue’s dominated convergence theorem.
The approach proposed below is valid whether n depends on j or not. This approach concerns the case where the random process X is a second-order, centred, WSS, continuous in quadratic mean, with PSD γ ∈ L∞([−π, π]). Therefore X is a band-limited random process, and we have (see proof in appendix D), Z
R
X(t)ΦS(t − k)dt = X[k]. (27) Consequently, US is the natural space of representation of such a random process. Note that according to (27), we can initialize the decomposition with the samples (X[k])k, by setting c0,0[k] = X[k], and calculating the wavelet packet coefficients with the recurrence described by relation (17). From now on, we will assume that the wavelet packet decomposition concerns the space US.
Before detailling this approach, the next subsection reminds the reader with a useful description of an arbitrary sequence (Wj,n)j≥1 of wavelet pack-ets by means of a binary sequence [16]. Then, in subsection 3.2, we treat the case where the quadrature mirror filters are the ideal low and high pass filters of the Shannon decomposition. The general case is addressed in subsection 3.3.
3.1
The binary sequence associated to a given sequence
of wavelet packets
Let κ = (`)` ∈ {0, 1}N
be an infinite binary sequence. For any natural number j, let nj(κ), in short nj, stand for the non negative integer
nj = nj(κ) = j X
`=1
`2j−` (28)
associated to the finite subsequence (`)`=1,2,...,j. Readily, nj is an element of Ij : nj = 0 when “ ` = 0 for all ` = 1, 2, · · · , j ”, and nj = 2j − 1 when “ ` = 1 for all ` = 1, 2, · · · , j ”.
Given an arbitrary infinite sequence κ = (`)` ∈ {0, 1}N
, the finite subse-quence (`)`=1,2,...,j, formed by the j first terms of κ, defines a unique non negative integer nj ∈ Ij and hence, is associated to the unique wavelet packet located at node (j, nj) of the decomposition tree. Moreover, this subsequence gives the unique path issued from US that leads to W
j,nj in
the wavelet packet tree. Basically, (`)`=1,2,...,j corresponds to the sequence H1, H2, . . . , Hj, of filters successively used to calculate the packet Wj,nj.
Conversely, let n ∈ Ij. There exists a unique finite sequence (`)`∈{1,2,...,j} of {0, 1}j such that n = nj where nj is given by (28).
From now on, given an infinite sequence κ, the natural number nj = n(κj) and the packet Wj,nj are said to be associated to each other. We also say
that the sequences κ and (Wj,nj)j are associated to each other.
Example 1: The following four sequences will often be used in the sequel to illustrate the results we present. These four sequences are
κ0 = (0, 0, 0, 0, 0, 0, · · · ), κ1 = (1, 0, 0, 0, 0, 0, · · ·), κ2 = (0, 1, 0, 0, 0, 0, · · · ), κ3 = (0, 0, 1, 0, 0, 0, · · ·),
In other words, the general term of the sequence κq, q = 0, 1, 2 and 3, is δ[q −`] for every natural number `. Clearly, we have that nj(κ0) = 0 for every natural number j. It is also very easy to see that, for q = 1, 2, 3, we have nj(κq) = 0 for j = 1, 2, . . . , q − 1, and that nj(κq) = 2j−q for j = q, q + 1, . . ..
3.2
Asymptotic decorrelation with the Shannon wavelet
packets
The QMF hS
0 and hS1 of the Shannon wavelet packet decomposition are the ideal low pass and high pass filters whose Fourier transforms HS
0 and H1S are: c HS 0(ω) = √ 2P`∈Zχ∆0(ω − 2π`) HS 1(ω) = √ 2P`∈Zχ∆1(ω − 2π`), (29) where ∆0 = −π 2, π 2 , and ∆1 = −π, −π 2] ∪ [ π 2, π
. Hereafter, we will use an upper index S in the notations of section 2 when the decomposition is achieved by using the ideal QMF HS
0 and H1S.
Let us define the map G by G(0) = 0 and the recurrence
G(2`) = 2G(`) if G(`) is even 2G(`) + 1 if G(`) is odd, (30) G(2` + 1) = 2G(`) + 1 if G(`) is even 2G(`) if G(`) is odd. (31) In a more compact form, we can write that,
G(2` + ) = 3G(`) + − 2 G(`) + 2 , (32)
where ∈ {0, 1} and bzc is the largest integer less than or equal to z. The map G is a permutation of N.
For every (k, `) ∈ N2, we define the sets ∆k,` by ∆k,` = −(` + 1)π2k , −`π2k ∪ `π 2k, (` + 1)π 2k . (33)
According to Coifman and Wickerhauser ( [14, p. 326], [17]), for every j > 0 and every n ∈ Ij, there exists a unique p = G(n) ∈ Ij such that
| dWj,n(ω)| = 2S j/2χ∆
j,p(ω), (34)
where, like in (6), WS
j,n(t) = 2−j/2WnS(2−jt), WnS stands for the map Wn recursively defined by (5) when the pair of QMF is (HS
0, H1S), and ∆j,p is given by (33). The set ∆j,p is the so-called subband. The wavelet packet decomposition of a signal of US, when this decomposition is based on the Shannon QMF, is the ideal subband coding of the signal under consideration.
The parameters of each subband can be selected using the integers j and p. The first one controls the bandwidth and the second one controls the central frequency.
For every j > 0, we have G(Ij) = Ij. Hence, the restriction of G to Ij is a permutation of Ij. This permutation makes it possible to re-order the wavelet packets WS
j,n from the lowest to the highest frequency supports. For further details on this re-ordering, the reader may refer to [14, p. 327].
Consider the sequence of wavelet packets (WSj,nj)j associated to an
ar-bitrary binary sequence κ in the sense given in subsection 3.1. At every resolution level j, it follows from (34) that, |W[S
j,nj| = 2 j/2χ∆ j,pj, with ∆j,pj = −(pj+ 1)π 2j , − pjπ 2j ∪ pjπ 2j , (pj + 1)π 2j (35)
and pj = G(nj). When j ranges over N, (35) defines a sequence (pjπ
2j )j. It is
easy to see that this sequence is Cauchy. Indeed, according to (9), we have nj+1 = 2nj+ , for some ∈ {0, 1}. Taking (32) into account, we can write that pj+1 = 2pj+ 0, for some 0 ∈ {0, 1}. With an easy recurrence, we obtain that pj+q = 2pj+q−1+ 0j+q = 2qpj + 2q−10j+1+ · · · + 210j+q−1+ 0j+q. Therefore, pj+qπ 2j+q − pjπ 2j = 1 2j q X `=1 0 `+j 2` ≤ 1 2j − 1 2j+q < 1 2j. As the sequence (pjπ
2j )j is Cauchy, it has a unique limit. This limit
a(κ) = lim j→+∞
pjπ
2j , (36)
will play a crucial role in the sequel. Note that 0 ≤ a(κ) ≤ π for every arbitrary sequence κ.
Example 1 (continued) : Consider the four sequences kq, q = 0, 1, 2, 3, introduced in example 1 above. For every natural number j, we have that pj(κ0) = G(nj(κ0)) = 0 since the general term of the sequence κ0 is zero.
The first q − 1 terms of the sequence κq when q = 1, 2, 3 are also 0. Therefore, we have pj(κq) = G(nj(κq)) = 0 for j = 1, 2, . . . , q − 1. Now, an easy recurrence shows that pj(κq) = G(nj(κq)) = 2j−q+1 − 1 for j = q, q + 1, . . ..
As far as the value of a(κq) is concerned for q = 0, 1, 2, 3, we easily see that limj→∞pj(κ0)/2j = 0, whereas limj→∞pj(κq)/2j = 1/2q−1 for q 6= 0. Therefore, we have a(κ0) = 0 and a(κq) = π/2q−1 for q = 1, 2 and 3.
We now state the following result.
Proposition 1 Let X be a second-order random process. Assume that X is centred, WSS, continuous in quadratic mean with PSD γ ∈ L∞([−π, π]).
Let κ = (k)k∈N be a binary sequence of {0, 1}N
. If a(κ) is a continuity point of γ, then lim j→+∞R S j,nj(κ)[m] = γ(a(κ))δ[m] (37)
uniformly in m ∈ Z, and where RS
j,nj(κ) is the autocorrelation function of the
wavelet packet coefficients of X with respect to WS j,nj.
Remark 3 The autocorrelation function RS
j,nj of the Shannon wavelet packet
coefficients cj,nj derives from (21) and, hence, is given by
Rj,nS j[m] = 1 2π Z R γ(ω)|W[S j,nj(ω)| 2ei2jmω dω. (38)
Proof: [of proposition 1]
The function γ is even because it is the Fourier transform of the even function R. As above, we write nj for nj(κ). From (34) and (38), we derive that RSj,nj[m] = 2 j π Z ∆+j,pj γ(ω) cos (2jmω)dω. (39) where ∆+j,pj = pjπ 2j , (pj + 1)π 2j . (40)
Let η > 0. Since γ is continuous at a(κ), there exists a positive real number α > 0, such that, for every ω ∈ [a(κ) − α, a(κ) + α], we have |γ(ω) − γ(a(κ))| < η.
In addition, it follows from (36), that
lim j→+∞pjπ2
−j = lim
j→+∞(pj + 1)π2
−j = a(κ), (41)
Thereby, there exists an integer j0 = j0(α), such that, for every natural number j ≥ j0, the values pjπ2−j and (pj + 1)π2−j are within the interval
[a(κ) − α, a(κ) + α]. It follows that, for every natural number j ≥ j0 and every ω ∈ ∆+
j,pj,
|γ(ω) − γ(a(κ))| < η. (42) Therefore, for all natural number j ≥ j0,
r2 j π Z ∆+j,pj |γ(ω) − γ(a(κ))| dω < η 2j π Z ∆+j,pj dω = η. (43)
On the other hand, we derive from (39) that for all natural number j ≥ j0, RSj,nj[m] − 2j π Z ∆+j,pj γ(a(κ)) cos (2jmω)dω = 2 j π Z ∆+j,pj(γ(ω) − γ(a(κ))) cos (2 jmω)dω ≤ 2 j π Z ∆+j,pj|γ(ω) − γ(a(κ))| dω. (44)
Hence, we derive from (43) and (44) that, for every natural number j ≥ j0 and every integer m,
|RS j,nj[m] − 2j π Z ∆+j,pj γ(a(κ)) cos (2jmω)dω| < η. (45) Since 2j π Z ∆+j,pj
γ(a(κ)) cos (2jmω)dω = γ(a(κ))δ[m], (46)
we conclude that, for every natural number j ≥ j0, RSj,nj[m] − γ(a(κ))δ[m] < η (47) uniformly in m ∈ Z.
Remark 4 Proposition 1 shows that the coefficients returned by the Shannon wavelet packet decomposition of the process X tend to be decorrelated when the resolution level tends to infinity.
Roughly speaking, the spectral measure of the discrete signal returned in a given subband by the Shannon wavelet packet decomposition of a WSS process tends, with increasing j, to γ(a(κ)).
This result also emphasizes the dependance between the autocorrelation function and the binary sequence κ. For instance, if we consider the four sequences κq, q = 0, 1, 2, 3 of exemple 1, we have that
lim j→+∞R
S
j,nj(κq)[m] = γ(π/2
q−1)δ[m] (48)
for q = 1, 2 and 3, whereas lim j→+∞R
S
j,nj(κ0)[m] = γ(0)δ[m]. (49)
Remark 5 (On the speed of the decorrelation process) Let ω0 > 0 and
γ(ω) = 1 − |ω| ω0 χ[−π,π]∩[−ω0,ω0](ω). (50)
We assume that γ represents the PSD of some centred WSS random process. We have Rj,nS j[m] = 2 j π Z ∆+j,pj∩[0,ω0] (1 − ω0ω ) cos (2jmω)dω. (51) If ω0 > π, ∆+j,pj ∩ [0, ω0] = ∆+j,pj, and we obtain that
rRSj,nj[m] = 1 − ωπ0 pj 2j − π ω0 1 2j+1 if m = 0 (−1)mpj−(−1)m(pj +1) πω0m22j if m 6= 0, (52) and thus, RSj,nj[m] − γ( πpj 2j )δ[m] ≤ ω0π 2j+11 . (53) It follows that γ(πpj/2j)δ[m] is an approximation of RS
j,nj[m] with a margin
of π/ω02j+1. This highlights the speed of the decorrelation process for ω0 > π. If ω0 ≤ π, then the function γ is null on [ω0, π]. If a(κ) is such that ω0< a(κ) ≤ π, then there exists α > 0 such that γ is null on ]a(κ) − α, a(κ) + α[ and there exists j1 such that for all j ≥ j1, ∆+j,pj ⊂ ]a(κ) − α, a(κ) + α[. Thus, the autocorrelation RS
j,nj[m] is null for all j ≥ j1. Now, consider
that 0 ≤ a(κ) < ω0. Then for j greater than or equal to a certain j2, ∆+j,pj ⊂ [0, ω0]. Thus, for all j ≥ j2, ∆+j,pj ∩ [0, ω0] = ∆
+
j,pj and we obtain that
(53) holds true. It is also easy to see that for j greater than or equal to a certain j3, (53) holds true when a(κ) = ω0. It follows that if ω0 ≤ π, there exists j0 such that for all j ≥ j0, (53) holds true.
More generally, consider an arbitrary PSD γ with support in [−π, π]. When the resolution level j is sufficiently large, γ can be seen as a linear function in ∆+j,pj. Then, we can generalise the above result, saying that there exists a finite level j0 such that for all resolution level j ≥ j0, RS
j,nj[m] is
approximatively equivalent to γ(πpj/2j)δ[m] with a margin of A/2j. The constant A and the level j0 depend on the shape of γ on ∆+j,pj.
3.3
Asymptotic decorrelation with non-ideal QMF
We consider QMF (h[r]0 , h[r]1 ) that depend on a parameter r such that, for ∈ {0, 1}, lim r→∞H [r] = HS (a.e), (54) where H0[r] and H [r]
1 are the Fourier transforms of h [r]
0 and h [r]
1 respectively, and where, as above, HS
0 and H1S are the Fourier transforms of the Shannon QMF. We assume that r is an integer. As mentioned in the introduction, this parameter is called the order of the QMF H0[r] and H1[r]. It has different meanings for the Daubechies and the Battle-Lemari´e QMF.
According to [18–20], the Daubechies QMF satisfy (54), the order r of a given pair (h[r]0 , h[r]1 ) of such QMF being the number of vanishing moments of the associated Daubechies wavelet function. The order r is also the zero multiplicity at π of H0[r].
It follows from [21] that the Battle-Lemari´e filters satisfy (54) as well; the order r of a given pair (h[r]0 , h[r]1 ) of such QMF is the spline order of the scaling function associated to H0[r].
Theorem 1 Let X be a second-order random process that satisfies the as-sumptions of proposition 1. Let κ = (k)k∈N be a binary sequence of {0, 1}N
. Consider, for every r, the sequence of wavelet packets (W[r]j,n
j(κ))j≥0associated
to κ, where the decomposition of US is achieved by the QMF (h[r] 0 , h
[r] 1 ). For every given natural number j, let R[r]j,n
j(κ) stands for the
autocorrela-tion funcautocorrela-tion of X with respect to the packet Wj,n[r]
j(κ). i) We have lim r→+∞R [r] j,nj[m] = R S j,nj[m], (55)
ii) In addition, if γ is continuous at a(κ), then, lim j→+∞ lim r→+∞R [r] j,nj(κ)[m] = γ(a(κ))δ[m] (56) uniformly in m ∈ Z.
Remark 6 According to theorem 1 and if γ is continuous at a(κ), then, for every given real number η > 0, there exists a natural number j0 with the following property: For every j ≥ j0, there exists r0 = r0(j, nj) > 0 such that, for every r ≥ r0 and every m ∈ Z,
R[r]j,nj(κ)[m] − γ(a(κ))δ[m] < η. (57) Thus, according to (57) we obtain nearly-white DWPT coefficients (with a margin of η) at resolution level j0 and by using QMF with order r0.
Proof: As above, we set nj = nj(κ). The autocorrelation function R[r] j,nj
derives from (21) and is equal to
Rj,n[r]j[m] = 1 2π
Z R
γ(ω)|W[j,n[r]j(ω)|2ei2jmωdω. (58) For every m ∈ Z, we have that,
R[r] j,nj [m] − R S j,nj[m] ≤ 2π1 Z R|γ(ω)| | [ Wj,n[r]j(ω)|2− |W[S j,nj(ω)| 2 dω. (59) But from appendix E, we have
d WS nj(ω) = " j Y `=1 HS`( ω 2j+1−`) # c ΦS(ω 2j), (60) and d Wn[r]j(ω) = " j Y `=1 H[r]`( ω 2j+1−`) # c ΦS(ω 2j). (61) Thus, from (6), (54), (60) and (61), and taking into acount that |H[r]`(ω)|
and |HS
`(ω)| are less than or equal to 1, we obtain that
lim r→+∞| [ Wj,n[r]j|2 = |W[S j,nj| 2 (a.e), (62)
and | [ Wj,n[r]j(ω)|2− |W[S j,nj(ω)| 2 ≤ 2j+1 cΦS(ω) 2. (63) Statement i) follows from (59), (62), (63), and Lebesgue’s dominated convergence theorem.
We just obtain statement ii) from proposition 1 and statement i).
4
Experimental results
The theoretical results presented above are of asymptotic nature. The role of this section is to reflect the manner in which the decorrelation appears in practice when the resolution level and the order of the filters are finite.
Consider a random process satisfying the assumptions of theorem 1. With the same notations as those used above, a consequence of theorem 1 is the existence of some resolution level j and some order r such that (57) holds true when the wavelet packet decomposition is initialized with the samples of the process. Equation (56) also tells us that the value of the autocorrelation function at lag 0 of the wavelet packet coefficients tend to the value of the PSD at a specific point that depends on the path followed in the wavelet packet tree. The following experimental results are aimed at illustrating these two facts.
We carried out experimental tests where the samples (X(k))k of the process X at the input of the wavelet packet decomposition form a dis-crete Auto-Regressive (AR) process. This AR process is such that X(k) = aX(k − 1) + W (k), where 0 < a < 1, and k = 1, 2, . . . , 220. The random vari-ables (W (k))k are Gaussian, independant and identically distributed with null mean and unitary standard-deviation. The samples (X(k))k were gen-erated by filtering the discrete random process (W (k))k through the discrete AR filter whose transfert function is (1 − a)/(1 − az−1). We present the re-sults obtained with a = 0.5 and a = 0.9. The corresponding PSDs are those displayed in figure 2, and the autocorrelations are displayed in figure 3.
The experimental results presented below show that, for the AR processes considered above, the asymptotic decorrelation stated by theorem 1 can ac-tually be attained with reasonable values for the resolution level j and the order r of the filters.
We begin with experimental results illustrating the influence of the res-olution level and the order of the QMF on the decorrelation process. We then address the convergence of the autocorrelation functions of the wavelet packet coefficients at lag 0.
0 0.5 1 1.5 2 2.5 3 3.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω γ(ω) a = 0.5 a = 0.9
Figure 2: PSDs of the random processes employed to carried out experimental tests. These random processes were synthesized by filtering some white noise with a first order AR filter.
−250 −20 −15 −10 −5 0 5 10 15 20 25 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t R(t) a = 0.5 a = 0.9
Figure 3: Autocorrelation functions of the random processes whose PSDs are plotted in figure 2.
4.1
Influence of the resolution level and the order of
the filters on the decorrelation process
We consider two full wavelet packet decompositions of the two random pro-cesses described above. These decompositions are obtained by using the Daubechies filters with orders 1 and 7. The empirical autocorrelation func-tions of the wavelet packet coefficients cj,nj(κq) for q = 0, 1, 2, 3, (see example
1) were calculated (we use 1500 points for every packet Wj,nj(κq)).
It turns out that the wavelet packet coefficients can be considered as significantly decorrelated when the resolution level is 6 and the order of the QMF is 7. For instance, figures 5 to 8 display the empirical normalized autocorrelation functions at resolution levels 3 and 6, when r equals 1 and 7 for the two AR processes. These autocorrelation functions are normalized in order to appreciate the gain in decorrelation. These figures underscore the importance of the role played by the resolution level and the order of the QMF.
The importance of the resolution level can be appreciated by noting that for the two processes, the coefficients at resolution level 6 are less correlated
than those obtained at level 3 (compare figure 5 and figure 6, then compare figure 7 to figure 8).
The role played by the QMF order is the following. At level 6 and for every wavelet packet, the decomposition based on the Daubechies QMF with order 7 yields a greater decorrelation than that obtained by the decomposition based on the Daubechies QMF with order 1 (in each of figures 6 and 8, compare the first and the second columns). This is not true at resolution level 3 (in each of figures 5 and 7, compare the first and the second columns). We can reasonably think that, at resolution level 3, we have not attained yet the resolution level j0 pointed out by theorem 1, and above which a nearly-decorrelation can be obtained if the QMF order is large enough.
4.2
The limit value of the correlation function at lag 0
Consider the sequence (R[r]j,nj)j where, for every natural number j, R[r]j,nj is the autocorrelation function of the wavelet packet coefficients associated to X with respect to the sequence (W[r]j,nj)j. According to theorem 1, the value R[r]j,nj(m) must be close to γ(a(κ))δ[m] if j and r are both large enough, r being above a value that depends on j. We recall that γ(a(κ)) is the value of the PSD of the random process X at a(κ) where a(κ) is given by (36) and κ is the binary sequence associated to (W[r]j,nj)j.
Let us illustrate this result for the value of the autocorrelation function at lag 0 and the binary sequences κ0, κ1, κ2, and κ3, introduced in example 1. We focus on the Daubechies QMF with order r = 7.
We still consider the two AR random processes introduced above and whose PSDs are those of figure 2. We observe that, quite rapidly, the terms of the sequence (R[7]j,nj[0])j becomes close to γ(a(κ)) when j increases. This shown in figure 9 and 10. These figures display the values R[7]j,n
j(κq)(0) for
j = 0, 1, 2, · · · , 9 and q = 0, 1, 2, 3. We plotted the limit value γ(a(κ)) in dotted line in these figures. We see that, as the resolution level j increases up to 6, the value at 0 of the autocorrelation function of the wavelet packet coefficients associated to each of the four sequences is sufficiently close to the value of the PSD at a(κ).
5
Conclusion
This paper provides further details concerning the asymptotic decorrelation of the wavelet packet coefficients issued from the decomposition of a band-limited WSS random process. By choosing a sufficiently large resolution level
and, then, by increasing the order of the filters with respect to the chosen resolution level, the wavelet packet coefficients tend to become decorrelated. The results presented in this paper complements those established in [10, 11, 15].
Figures 5 to 8 highlight the quality of the decorrelation process. This quality can be regarded as rather good: Compare the autocorrelation func-tions of the input WSS processes (figure 3) to the autocorrelation funcfunc-tions of the associated coefficients at level 6 (figure 6 and figure 8). The perfor-mance of this experiment can be improved by increasing the order of the QMF (compare the first and the second column of figure 8). These results suggest using a full wavelet packet decomposition with six or more resolu-tion levels and Daubechies filters with seven or more vanishing moments. The experimental results presented in this paper indicate that the asymp-totic decorrelation can be attained in practice with reasonable values for the resolution level and the order of the QMF. This relates to the convergence speed of the Daubechies QMF to the Shannon filters (see figure 4) and also to the speed of the decorrelation process (see remark 5).
0 0.5 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ω H0(ω) db1 db2 db4 db7 db20 db40
Figure 4: Magnitude response of some Daubechies scaling filters. This figure illustrates the convergence speed of the Daubechies QMF to the Shannon filters. It displays the magnitude response of the Daubechies scaling filters with orders 1, 2, 4, 7, 20, and 40: dbr stands for the Daubechies scaling filter of order r. This scaling filter is obtained by using the matlab routine dbaux of the wavelet toolbox.
R3,n[1]3(κ0)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]3,n3(κ0)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R3,n[1]3(κ1)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]3,n3(κ1)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R3,n[1]3(κ2)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]3,n3(κ2)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R3,n[1]3(κ3)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]3,n3(κ3)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m
Figure 5: Autocorrelation functions of the wavelet packet coefficients re-turned by the decomposition of the random process whose PSD is that of figure 2, with a = 0.5. The resolution level is j = 3. The first column displays the results obtained by using the Daubechies filters with order 1. The second column displays the results obtained by using the Daubechies filters with or-der 7. Each line corresponds to one of the four sequences κq, q = 0, 1, 2, and 3 considered in example 1. The autocorrelation functions are normalized for a better appreciation of the difference between the autocorrelation functions.
R6,n[1]6(κ0)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]6,n6(κ0)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R6,n[1]6(κ1)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]6,n6(κ1)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R6,n[1]6(κ2)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]6,n6(κ2)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R6,n[1]6(κ3)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]6,n6(κ3)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m
Figure 6: Autocorrelation functions of the wavelet packet coefficients re-turned by the decomposition of the random process whose PSD is that of figure 2, with a = 0.5. The resolution level is j = 6. The first column displays the results obtained by using the Daubechies filters with order 1. The second column displays the results obtained by using the Daubechies filters with or-der 7. Each line corresponds to one of the four sequences κq, q = 0, 1, 2, and 3 considered in example 1. The autocorrelation functions are normalized for a better appreciation of the difference between the autocorrelation functions.
R3,n[1]3(κ0)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]3,n3(κ0)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R3,n[1]3(κ1)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]3,n3(κ1)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R3,n[1]3(κ2)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]3,n3(κ2)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R3,n[1]3(κ3)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]3,n3(κ3)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 m
Figure 7: Autocorrelation functions of the wavelet packet coefficients re-turned by the decomposition of the random process whose PSD is that of figure 2, with a = 0.9. The resolution level is j = 3. The first column displays the results obtained by using the Daubechies filters with order 1. The second column displays the results obtained by using the Daubechies filters with or-der 7. Each line corresponds to one of the four sequences κq, q = 0, 1, 2, and 3 considered in example 1. The autocorrelation functions are normalized for a better appreciation of the difference between the autocorrelation functions.
R6,n[1]6(κ0)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]6,n6(κ0)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R6,n[1]6(κ1)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]6,n6(κ1)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R6,n[1]6(κ2)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]6,n6(κ2)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R6,n[1]6(κ3)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m R[7]6,n6(κ3)[m] −25 −20 −15 −10 −5 0 5 10 15 20 25 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 m
Figure 8: Autocorrelation functions of the wavelet packet coefficients re-turned by the decomposition of the random process whose PSD is that of figure 2, with a = 0.9. The resolution level is j = 6. The first column displays the results obtained by using the Daubechies filters with order 1. The second column displays the results obtained by using the Daubechies filters with or-der 7. Each line corresponds to one of the four sequences κq, q = 0, 1, 2, and 3 considered in example 1. The autocorrelation functions are normalized for a better appreciation of the difference between the autocorrelation functions.
R[7]j,n j(κ0)[0] 0 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 j R[7]j,nj(κ1)[0] 0 1 2 3 4 5 6 7 8 9 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 j R[7]j,n j(κ2)[0] 0 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 j R[7]j,n j(κ3)[0] 0 1 2 3 4 5 6 7 8 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 j
Figure 9: Convergence of R[7]j,nj(κq)[0] to γ(a(κq)) with j, for the four test sequences κq introduced in example 1 for q = 0, 1, 2, 3. The decomposition concerns the random process whose PSD is that of figure 2, with a = 0.5, and was achieved by using the Daubechies filters with order 7. The values of a(κq) and γ(a(κq)), q = 0, 1, 2, 3, are given in example 1 (continued) and remark 4. As a matter of fact, we have that a(κ0) = 0 and a(κq) = π/2q−1 for q = 1, 2 and 3. The dotted lines on these figures represent the value of γ(a(κq)) for q = 0, 1, 2, 3.
R[7]j,n j(κ0)[0] 0 1 2 3 4 5 6 7 8 9 0 0.2 0.4 0.6 0.8 1 1.2 1.4 j R[7]j,nj(κ1)[0] 0 1 2 3 4 5 6 7 8 9 0 0.01 0.02 0.03 0.04 0.05 0.06 j R[7]j,n j(κ2)[0] 0 1 2 3 4 5 6 7 8 9 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 j R[7]j,n j(κ3)[0] 0 1 2 3 4 5 6 7 8 9 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 j
Figure 10: Convergence of Rj,n[7]j(κq)[0] to γ(a(κq)) with j for the four test sequences κq introduced in example 1 for q = 0, 1, 2, 3. The decomposition concerns the random process whose PSD is that of figure 2, with a = 0.9, and was achieved by using the Daubechies filters with order 7. The values of a(κq) and γ(a(κq)), q = 0, 1, 2, 3, are given in example 1 (continued) and remark 4. As a matter of fact, we have that a(κ0) = 0 and a(κq) = π/2q−1 for q = 1, 2 and 3. The dotted lines on these figures represent the value of γ(a(κq)) for q = 0, 1, 2, 3.
Appendix A
According to Loeve’s lemma ( [22]), and for −∞ < a ≤ b < ∞, the stochastic
integral Z
b a
X(t)Wj,n,k(t)dt (64)
is well defined if, and only if
E Z b a X(t)Wj,n,k(t)dt Z b a X(s)Wj,n,k(s)ds (65)
is well defined, i.e. if, and only if the Riemann integral Z b
a Z b
a
R(t, s)Wj,n,k(t)Wj,n,k(s)dtds (66)
exist and is finite. We can then define the stochastic integral Z
R
X(t)Wj,n,k(t)dt (67)
as the limit ofRabX(t)Wj,n,k(t)dt (with convergence in quadratic mean) when a → −∞ and b → +∞, and this, provided that the integral
ZZ R2
R(t, s)Wj,n,k(t)Wj,n,k(s)dtds (68)
exists. This is the case if we suppose that Wnis compactly supported or has a sufficiently fast decay (see [6] among others). Given a resolution level j and n ∈ Ij (so, given Wj,n), the wavelet packet decomposition of X returns the random variables
cj,n[k] = Z
R
X(t)Wj,n,k(t)dt, k ∈ Z. (69)
Appendix B
According to (5), (6) and (7), and for ∈ {0, 1}, we have Wj+1,2n+,k(t) = τ2j+1kWj+1,2n+(t) = τ2j+1k2−(j+1)/2W2n+(2−(j+1)t) = τ2j+1k2−(j+1)/2 √ 2X `∈Z h[`]Wn(2−jt − `)
= τ2j+1k2−j/2 X `∈Z h[`]Wn(2−jt − `) = 2−j/2X `∈Z h[`]Wn(2−jt − 2k − `) = 2−j/2X `∈Z h[` − 2k]Wn(2−jt − `) = X `∈Z h[` − 2k]Wj,n,`(t), (70) and thus Wj+1,2n+,k = P`∈Zh[` − 2k]Wj,n,`. The equalities above must be understood in the quadratic sense. If the impulse response of the filters have finite length, these equalities hold true pointwise.
Appendix C
With the notations of appendix A, we assume that the mean-square integral (64) is well defined. The autocorrelation function Rj,nof the discrete random process cj,n is defined by
Rj,n[k, `] = ZZ
R2R(t − s)W
j,n,k(t)Wj,n,`(s)dtds, (71)
which can be written in the following form
Rj,n[k, `] = ZZ
R2
R(t)Wj,n,k(t + s)Wj,n,`(s)dtds. (72)
In addition, since Wj,n,k(t) = 2−j/2Wn(2−jt − k), we have Z R Wj,n,k(t + s)Wj,n,`(s)ds, = Z R Wn(s + 2−jt − k)Wn(s − `)ds, = hτk−`−2−jtWn, WniL2(R), = 1 2π Z R | cWn(ω)|2ei(k−`−2−jt)ω dω, (73)
where we used the Parseval formula to obtain the last equality. We thus have
rRj,n[k, `] = 1 2π Z Z R2R(t)| c Wn(ω)|2ei(k−`−2−jt)ω dtdω. (74)
And since Z R R(t)e−i2−jtω dt = γ(ω 2j), (75) we derive that Rj,n[k, `] = 1 2π Z R γ(ω 2j)| cWn(ω)| 2ei(k−`)ωdω. (76)
Appendix D
Let Z be the orthogonal stochastic measure, associated with the spectral measure m of the continuous and second order WSS random process X (Wiener-Khintchine’s theorem). We have that
m(dω) = 1
2πγ(ω)dω, (77) where γ is the PSD of X, γ has support in [−π, π]. Then, we can write X(t) as a stochastic integral :
X(t) = Z
R
eitωZ(dω). (78)
Now, according to [23, lemma 5, p. 197-198], we have Z R X(t)ΦS(t − k)dt = Z R Z R eitωZ(dω) ΦS(t − k)dt, = Z R Z R ΦS(t − k)eitωdt Z(dω), = Z R c ΦS(ω)eikωZ(dω). (79) We obtain that X(k) − Z R X(t)ΦS(t − k)dt = Z R eikω(1 − cΦS(ω))Z(dω), (80) and then, E " X(k) − Z R X(t)ΦS(t − k)dt 2# = Z R 1 − cΦS(ω) 2m(dω). = 1 2π Z π −π 1 − cΦS(ω) 2γ(ω)dω. (81)
Taking into account the fact that cΦS = χ [−π,π], we derive that E " X(k) − Z R X(t)ΦS(t − k)dt 2# = 0, (82)
and thus, that
X(k) = Z R X(t)ΦS(t − k)dt (83) in quadratic mean.
Appendix E
Lemma 1 With the same notations as in section 2.1, and for n = Pj`=1`2j−`, we have that c Wn(ω) = " j Y `=1 m`( ω 2j+1−`) # b Φ(ω 2j). (84)
Proof: According to (5), for ∈ {0, 1}, we have \ W2n+(ω) = m( ω 2) cWn( ω 2). (85)
By taking of account (28), we obtain successively c Wn(ω) = mj( ω 2) \Wn−j2 (ω 2), = mj( ω 2)mj−1( ω 22) · · · m1( ω 2j)bΦ( ω 2j), (86) and thus c Wn(ω) = " j Y `=1 m`( ω 2j+1−`) # b Φ(ω 2j). (87)
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![Figure 9: Convergence of R [7] j,n j (κ q ) [0] to γ(a(κ q )) with j, for the four test sequences κ q introduced in example 1 for q = 0, 1, 2, 3](https://thumb-eu.123doks.com/thumbv2/123doknet/8121692.272556/30.918.288.585.110.813/figure-convergence-r-κ-test-sequences-introduced-example.webp)
![Figure 10: Convergence of R j,n [7] j (κ q ) [0] to γ(a(κ q )) with j for the four test sequences κ q introduced in example 1 for q = 0, 1, 2, 3](https://thumb-eu.123doks.com/thumbv2/123doknet/8121692.272556/31.918.288.585.112.814/figure-convergence-r-κ-test-sequences-introduced-example.webp)
